L(s) = 1 | + (0.866 − 0.5i)2-s + (−1.65 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.18 + 1.26i)6-s + (−2.92 + 1.68i)7-s − 0.999i·8-s + (2.5 − 1.65i)9-s + (2.18 + 3.78i)11-s + (−0.396 + 1.68i)12-s + (5.84 + 3.37i)13-s + (−1.68 + 2.92i)14-s + (−0.5 − 0.866i)16-s + 1.62i·17-s + (1.33 − 2.68i)18-s + 2.37·19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.957 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.484 + 0.515i)6-s + (−1.10 + 0.637i)7-s − 0.353i·8-s + (0.833 − 0.552i)9-s + (0.659 + 1.14i)11-s + (−0.114 + 0.486i)12-s + (1.61 + 0.935i)13-s + (−0.450 + 0.780i)14-s + (−0.125 − 0.216i)16-s + 0.394i·17-s + (0.314 − 0.633i)18-s + 0.544·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26649 + 0.462498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26649 + 0.462498i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (1.65 - 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2.92 - 1.68i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.84 - 3.37i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.62iT - 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + (1.18 + 0.686i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.686 - 1.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.37 - 4.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.87 - 2.81i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.38 + 3.68i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-2.18 + 3.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.05 - 7.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 3.11iT - 73T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.38 + 3.68i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + (-7.25 + 4.18i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.41865107727005738868380586834, −10.38256769238452279654714893748, −9.628960007593209203679803113919, −8.825090374263054551131909471982, −6.91246098437853349122661475642, −6.41561056692283734421664266781, −5.52218628704808051719848505710, −4.31085748173774152663799254705, −3.46702681519771797347501203011, −1.61421667760178960379554091417,
0.863790399164499406792235629297, 3.25912035330846372519807495399, 4.09249723866947438986115663144, 5.69701443427365082483110096184, 6.06950134094479013715043157076, 6.98749613966837751395243806365, 7.975330966373735560570685583742, 9.199575087057147277127316527227, 10.41102417374131925947521100779, 11.09888539479555043397950344438